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<\body>
  <doc-data|<doc-title|2023-06-02>>

  \<#95EE\>\<#9898\>: \<#5728\>\<#96C6\>\<#5408\><math|<around*|{|1,2,\<ldots\>,2000|}>>\<#7684\>\<#6240\>\<#6709\>\<#5B50\>\<#96C6\>\<#4E2D\>\<#FF0C\>\<#6EE1\>\<#8DB3\>\<#5143\>\<#7D20\>\<#4E4B\>\<#548C\>\<#80FD\>\<#88AB\>5\<#6574\>\<#9664\>\<#7684\>\<#5B50\>\<#96C6\>\<#6709\>\<#591A\>\<#5C11\>\<#4E2A\>?

  \<#89E3\>\<#7B54\>: \<#4F5C\>\<#591A\>\<#9879\>\<#5F0F\>

  <\equation*>
    f<around*|(|x|)>=<around*|(|1+x|)><around*|(|1+x<rsup|2>|)>\<cdots\><around*|(|1+x<rsup|2000>|)>
  </equation*>

  \<#5219\>\<#5176\>\<#5C55\>\<#5F00\>\<#4E4B\>\<#540E\>\<#FF0C\>\<#6B21\>\<#6570\>\<#662F\>5\<#7684\>\<#500D\>\<#6570\>\<#7684\>\<#90A3\>\<#4E9B\>\<#9879\>\<#7684\>\<#7CFB\>\<#6570\>\<#4E4B\>\<#548C\>\<#5373\>\<#4E3A\>\<#6240\>\<#6C42\>(\<#5305\>\<#62EC\>\<#5E38\>\<#6570\>\<#9879\>\<#FF0C\>\<#5B83\>\<#5BF9\>\<#5E94\>\<#7684\>\<#5B50\>\<#96C6\>\<#4E3A\>\<#7A7A\>\<#96C6\>).

  \<#6B64\>\<#591A\>\<#9879\>\<#5F0F\>\<#7684\>\<#6B21\>\<#6570\>\<#662F\>
  <math|1+2+\<cdots\>+2000=2001000>, \<#8BBE\>\<#5176\>\<#5C55\>\<#5F00\>\<#5F0F\>\<#4E3A\>

  <\equation*>
    f<around*|(|x|)>=a<rsub|0>+a<rsub|1>x+\<cdots\>+a<rsub|2001000>x<rsup|2001000>
  </equation*>

  \<#4E0B\>\<#9762\>\<#6C42\> <math|a<rsub|0>+a<rsub|5>+a<rsub|10>+\<cdots\>+a<rsub|2001000>>
  \<#7684\>\<#503C\>.

  \<#8BBE\> <math|\<zeta\><rsub|k><around*|(|k=0,1,2,3,4|)>>
  \<#4E3A\>\<#4E94\>\<#6B21\>\<#5355\>\<#4F4D\>\<#6839\>\<#FF0C\>\<#5176\>\<#4E2D\>
  <math|\<zeta\><rsub|k>=e<rsup|i<frac|2k\<pi\>|5>>=cos<frac|2k\<pi\>|5>+i
  sin<frac|2k\<pi\>|5><around*|(|k=0,1,2,3,4|)>>\<#FF0C\>\<#6B64\>\<#5904\>
  <math|i> \<#4E3A\>\<#865A\>\<#6570\>\<#5355\>\<#4F4D\>\<#FF0C\>\<#5219\>\<#6709\>
  <math|\<zeta\><rsub|0>=1> \<#4E14\> <math|\<zeta\><rsub|k>=\<zeta\><rsub|1><rsup|k>>.
  \<#6709\>

  <\equation*>
    <big|sum><rsub|k=0><rsup|4>f<around*|(|\<zeta\><rsub|k>|)>=5a<rsub|0>+a<rsub|1><big|sum><rsub|k=0><rsup|4>\<zeta\><rsub|k>+a<rsub|2><big|sum><rsub|k=0><rsup|4>\<zeta\><rsub|k><rsup|2>+\<cdots\>+a<rsub|2001000><big|sum><rsub|k=0><rsup|4>\<zeta\><rsub|k><rsup|2001000>
  </equation*>

  \<#8003\>\<#5BDF\>\<#548C\>\<#5F0F\> <math|<big|sum><rsub|k=0><rsup|4>\<zeta\><rsub|k><rsup|m><around*|(|m\<in\>N|)>>
  \<#7684\>\<#53D6\>\<#503C\>\<#60C5\>\<#51B5\>\<#FF0C\>\<#7531\>\<#5355\>\<#4F4D\>\<#6839\>\<#6027\>\<#8D28\>\<#FF0C\>\<#6613\>\<#77E5\>

  <\equation*>
    <big|sum><rsub|k=0><rsup|4>\<zeta\><rsub|k><rsup|m>=<choice|<tformat|<table|<row|<cell|0>|<cell|5\<nmid\>m>>|<row|<cell|5>|<cell|5\<mid\>m>>>>>
  </equation*>

  \<#56E0\>\<#6B64\>

  <\equation*>
    <big|sum><rsub|k=0><rsup|4>f<around*|(|\<zeta\><rsub|k>|)>=5<around*|(|a<rsub|0>+a<rsub|5>+\<cdots\>+a<rsub|2001000>|)>
  </equation*>

  \<#6240\>\<#4EE5\>\<#9898\>\<#76EE\>\<#7684\>\<#7B54\>\<#6848\>\<#4FBF\>\<#662F\>
  <math|<frac|1|5><big|sum><rsub|k=0><rsup|4>f<around*|(|\<zeta\><rsub|k>|)>>\<#FF0C\>\<#4E0B\>\<#9762\>\<#8BBE\>\<#6CD5\>\<#6C42\>\<#51FA\>\<#6B64\>\<#503C\>\<#FF0C\>\<#9996\>\<#5148\>\<#6709\>
  <math|f<around*|(|\<zeta\><rsub|0>|)>=f<around*|(|1|)>=2<rsup|2000>>\<#FF0C\>\<#800C\>\<#5BF9\>
  <math|f<around*|(|\<zeta\><rsub|k>|)><around*|(|k=1,2,3,4|)>> \<#6709\>

  <\eqnarray*>
    <tformat|<table|<row|<cell|f<around*|(|\<zeta\><rsub|k>|)>>|<cell|=>|<cell|<around*|(|1+\<zeta\><rsub|k>|)><around*|(|1+\<zeta\><rsub|k><rsup|2>|)>\<cdots\><around*|(|1+\<zeta\><rsub|k><rsup|2000>|)>>>|<row|<cell|>|<cell|=>|<cell|<around*|[|2<around*|(|1+\<zeta\><rsub|1>|)><around*|(|1+\<zeta\><rsub|2>|)><around*|(|1+\<zeta\><rsub|3>|)><around*|(|1+\<zeta\><rsub|4>|)>|]><rsup|400>>>|<row|<cell|>|<cell|=>|<cell|<around*|[|2\<cdot\><around*|(|1+\<zeta\><rsub|1>|)><around*|(|1+\<zeta\><rsub|1><rsup|2>|)><around*|(|1+\<zeta\><rsub|1><rsup|3>|)><around*|(|1+\<zeta\><rsub|1><rsup|4>|)>|]><rsup|400>>>|<row|<cell|>|<cell|=>|<cell|<around*|[|2\<cdot\><frac|1-\<zeta\><rsub|1><rsup|8>|1-\<zeta\><rsub|1>>\<cdot\><around*|(|1+\<zeta\><rsub|1><rsup|3>|)>|]><rsup|400>>>|<row|<cell|>|<cell|=>|<cell|<around*|[|2\<cdot\><frac|1-\<zeta\><rsub|1><rsup|6>|1-\<zeta\><rsub|1>>|]><rsup|400>>>|<row|<cell|>|<cell|=>|<cell|2<rsup|400>>>>>
  </eqnarray*>

  \<#6240\>\<#4EE5\>\<#6700\>\<#7EC8\>\<#7684\>\<#7B54\>\<#6848\>\<#662F\>
  <math|<frac|1|5><around*|(|2<rsup|2000>+4\<times\>2<rsup|400>|)>>.

  \;

  \;

  \;

  \;

  \;

  \;

  \;

  \;
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